The average of the values of a function at random points

The behavior of (1/N) asN→∞ is considered, wheref is a bounded measurable function on (−∞, ∞) and (S _n) _n ^=1/∞ are the partial sums of a sequence of independent and identically distributed rondom variables.     

Sampling of Paley-Wiener functions on stratified groups

We consider a generalization of entire functions of spherical exponential type on stratified groups. An analog of the Paley-Wiener theorem is given. We also show that every spectral entire function on a stratified group is uniquely determined by its values on some discrete subgroups. The main result of the article is reconstruction formula of spectral entire functions from their values on discrete subgroups using Lagrangian splines.     

Chaotic states in a random world: Relationship between the nonlinear differential equations of excitability and the stochastic properties of ion channels

Excitable membranes allow cells to generate and propagate electrical signals. In the nervous system these signals transmit information, in muscle they trigger contraction, and in heart they regulate spontaneous beating. A central question in excitability theory concerns the relationship between the aggregate properties of membranes (marcoscopic) and the properties of channels in the membranes (mircoscopic). Hodgkin and Huxley (1952) laid the foundations of membrane excitability, and Neher and Sakmann (1976) developed techniques to study individual channels. This article focuses on the relationship between the macroscopic domain, in which non-linear differential equations determine the electrical properties of cells, and the microscopic domain, in which the probabilistic nature of channels establishes the pattern of activity. Using nerve cell membranes as an example, we examine how information in one domain predicts behavior in the other. We conclude that the probabilistic nature of channels generates virtually all macroscopic electrical properties, including resting potentials, action potentials, spontaneous firing, and chaos.     

Spherical functions and integral geometry

The paper presents two proofs of an integral geometric formula concerningn-dimensional ellipsoids. One of the proofs is based on a representation theorem for spherical functions due to Harish-Chandra.     

Synthesis of substituted tetrahydropyrimido-[4, 5-b][1, 7]naphthyridines as potential antitumor agents

The synthesis of two linear trisubstituted 6, 7, 8, 9-tetrahydropyrimido[4, 5-b][1, 7]naphthyridines 1 and 3 were accomplished by the regiospecific cyclocondensation of two disubstituted 6-aminopyrimidines 10 and 11 with the ketoaldehyde 1-benzyl-4-hydroxymethylene-3-piperidone. Catalytic debenzylation of 1 afforded the disubstituted compound 2 . The linear structures of 1–3 were established by ¹H nmr and 13C nmr.     

The chemistry of 5-hydroxy-6-(hydroxyalkyl)uracils. Synthesis of spiro[pyrimidine-4,2′-pyrano[3,2-d]pyrimidines]

Both 5-acetoxy-6-(acetoxymethyl)-1-methyluracil 1 and its parent diol 11 are converted into the spiro[pyrimidine-4,2′-pyrano[3,2-d]pyrimidine] 6 when treated, respectively, with hot methanolic pyridine and with one equivalent of acetic anhydride. The formation of 6 can be explained in terms of the generation and dimerization of the reactive 5-oxo-6-methylene pyrimidine 2 . The structure of 6 was determined by ¹³C nmr spectroscopy and by chemical transformations that lead to the pyrimidinylethylhydantoin 9 and the 6,6′-[1,2-ethanediyl]bispyrimidine 10 . The more complex 5-hydroxy-6-(hydroxyalkyl)uracils represented by the 6,5′-cyclourid-ines 17 undergo an analogous dimerization when treated with acetic anhydride to give structures 22a and 22b . Dimer 22a was also prepared via the 5-phosphate ester 18 . The stereochemistry of dimers 22a and 22b , which is apparent from their ¹H nmr spectra, indicates that two molecules of the enones 21a or 21b dimerize in a highly stereoselective manner.     

EFFECT OF CHRONIC UV EXPOSURE ON EPIDERMAL FORWARD SCATTERING-ABSORPTION INSK–1 HAIRLESS MOUSE SKIN

Abstract— Clinical and histological precancerous responses to UV irradiation are complicated dynamic functions of total dose, dose fractionation, fluence rate, and spectral distribution. This may be due, in large part, to the ability of UV to decrease epidermal-stratum corneum transmission by stimulation of hyperplasia. This work provides quantitative measurement of dose- and wavelength-dependent optical changes inSK–1 hairless mouse epidermis-stratum corneum occurring under irradiation with “monochromatic” UV wavebands, at 280, 290, 300, 307, and 313 nm. Mice were irradiated 5 days per week with a filtered Xenon-Hg high-intensity grating monochromator, starting with 0.9 minimal erythemal dose (MED), followed by incremental increases in the radiation dose by 20% of the original dose every tenth irradiation day, for2–8 consecutive weeks. Subsequent irradiations (for longer experiments) were followed by 30% incremental increases after the 8th week every 10th irradiation day until cessation of radiation at the end of 14 weeks. Irradiated and control full-thickness epidermis/ stratum corneum were examined histologically and by forward-scattering absorption spectroscopy. Chronic irradiation of hairless mice resulted in significant hyperplasia which was optically manifested by a general increase in forward-scattering absorbance. At moderate local doses (7.2 MED), the absorbance increase per MED was approximately the same for all excitation wavelengths, whereas at large total doses (? 100 MED) the optical increase per delivered MED progressively decreased in the order 313> 307> 300? 290> 280 nm. The increase in skin thickening, expressed as observed increase in absorption at 320 nm, correlated well with histological and clinical data. We propose that optical changes induced by UV-induced thickening can account in large part, if not entirely, for dynamic changes in action spectra for (pre) cancerous processes under chronic irradiation conditions.     

Recent efforts to increase greenness in chromatography

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